Page 98 - Curvature and Homology

80 11. TOPOLOGY OF DIFFERENTIABLE MANIFOLDS

7. Define themapp:AP(V*)-tAn-P(V*) by p=hoy, wherey: AP(V*)
+ (V) is the natural identification map determined by the inner product
in AP(V*). Then, p is the star operation of Hodge.
8. Let V be a vector space (over R) with the properties:
(i) V is the direct sum of subspaces Vp where p runs through non-negative
integers and
(ii) V has a coboudary operator that is an endomorphism d of V such that
d,Vv C V9+l with d,,,d, = 0 where d, denotes the restriction of d to Vr.
The vector space
kernel d,
Hp(V) =
image d,,
is called the p* cohomology vector space (or group) of V. A theory based on V
together with the operator d is usually called a cohomology theory or d-cohomology
theory when emphasis on the coboundary operator is required. We have seen
that the Grassman algebra A(T*) with the exterior differential operator d gives
rise to the de Rharn cohomology theory. On the other hand, a cohomology
theory is defined by the pair (A(T*),6) on a Riemannian manifold by setting
A-9 = A 9, p = 0,1,2, *-* . Prove that the * operator induces an isomorphism
between the two cohomology theories.

B. The operators H and G on a compact manifold
1. Show that for any a E AP(T*) there exists a unique p-form H[a] in A UT*)
with the property (a#) = (H[or],/3) for all /3 E A T*).
2. Prove that H[H[u]] = H[a] for anyp-form a.
3. For a given p-form a there exists a p-form /3 satisfying the differential
equation A/3 = a - H[a]. Show that any two solutions differ by a harmonic
p-form and thereby establish the existence of a unique solution orthogonal to
A&(T*). Denote this solution by Ga and show that it is charaderized by the
conditions
a = AGa + H[a] and (Ga,/3) = 0
for any E AHT*).
The operator G is called the Green's operator.
4. Prove that H[Ga] vanishes for any p-form a.
5. Prove:
(a) The operators H and G commute with d, 6, A and *;
(b) G is sey-dud, that is
(Ga,B) = (a,G/3)
for any a, /3 of degree P;